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Theorem atn0 29498
Description: An atom is not zero. (atne0 22925 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atne0.z  |-  .0.  =  ( 0. `  K )
atne0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atn0  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )

Proof of Theorem atn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 atne0.z . . . 4  |-  .0.  =  ( 0. `  K )
4 atne0.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat3 29497 . . 3  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  ( Base `  K
)  /\  P  =/=  .0.  /\  A. x  e.  ( Base `  K
) ( x ( le `  K ) P  ->  ( x  =  P  \/  x  =  .0.  ) ) ) ) )
6 simp2 956 . . 3  |-  ( ( P  e.  ( Base `  K )  /\  P  =/=  .0.  /\  A. x  e.  ( Base `  K
) ( x ( le `  K ) P  ->  ( x  =  P  \/  x  =  .0.  ) ) )  ->  P  =/=  .0.  )
75, 6syl6bi 219 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  ->  P  =/= 
.0.  ) )
87imp 418 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atncvrN  29505  atnle  29507  atlatmstc  29509  intnatN  29596  atcvrneN  29619  atcvrj2b  29621  2llnm3N  29758  pmapjat1  30042  lhpocnle  30205  lhpmatb  30220  lhp2atnle  30222  trlatn0  30361  ltrnnidn  30363  trlnidatb  30366  cdlemg33c  30897  cdlemg33e  30899  dihatexv  31528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-undef 6298  df-riota 6304  df-plt 14092  df-glb 14109  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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